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Beauville surfaces with abelian Beauville group

A Beauville surface is a rigid surface of general type arising as a quotient of a product of curves $C_{1}$, $C_{2}$ of genera $g_{1},g_{2}\ge 2$ by the free action of a finite group $G$. In this paper we study those Beauville surfaces for which $G$ is abelian (so that $G\cong \mathbb{Z}_{n}^{2}$ with $\gcd(n,6)=1$ by a result of Catanese). For each such $n$ we are able to describe all such surfaces, give a formula for the number of their isomorphism classes and identify their possible automorphism groups. This explicit description also allows us to observe that such surfaces are all defined over $\mathbb{Q}$.